Arc Reduction in Timed Arc Petri Nets

• Arc reduction is a family of methods that simplifies complex models by eliminating redundant arcs through domain-specific insights like symmetries and cost structures.
• It employs a refined token-inclusion scheme in timed-arc Petri nets to safely prune the state space, thereby accelerating forward reachability and verification.
• The approach maintains essential system behavior by carefully handling monotonicity-breaking features such as inhibitor arcs and age invariants.

The arc reduction approach encompasses a family of methodologies designed to decrease the computational complexity, dimensionality, or size of models and data structures in systems involving arcs, ranging from Petri nets and transmission grids to radiative heat transfer models and combinatorial optimization. These methods leverage domain-specific insights—such as monotonicity properties, symmetries, cost structures, or connectivity—in order to safely eliminate redundant or non-influential arcs (or tokens, branches, or connections), accelerating computation and facilitating analysis without sacrificing essential system features or correctness.

1. Refined Token-Inclusion in Timed Arc Petri Nets

In the context of bounded timed-arc Petri nets (TAPNs), the arc reduction technique centers on a symbolic marking inclusion relation that is robust under monotonicity-breaking features (age invariants, inhibitor arcs). Instead of classical subset inclusion, the approach partitions tokens into eligible/ineligible categories by place invariants, inhibitor arcs, and dynamic age bounds. Tokens whose age surpasses the maximum constant for a place are treated as indistinguishable, enabling symmetry-based reductions. The formal ordering $M \sqsubseteq M'$ between symbolic markings $M = (pl, W)$ and $M' = (pl', W')$ holds if there exists a bijection $h: \mathcal{I}(M) \to \mathcal{I}(M')$ matching discrete places, and for every valuation $v \in W$ there exists $v' \in W'$ matching ages exactly or both exceeding the place’s age bound. This inclusion ordering is maintained except in the presence of monotonicity-breaking properties (such as token count tests), ensuring safe state-space pruning.

2. State-Space Reduction and Symmetry Exploitation

The refined arc reduction supports substantial pruning of the symbolic state space during forward reachability analysis. Once a marking $M$ is proven to be included in $M'$, all future behaviors stemming from $M$ are subsumed by $M'$. This allows the verification algorithm (implemented in TAPAAL) to discard already seen or successor markings when larger covering markings are present. Coupled with symmetry reduction and extrapolation, this leads to compact state spaces highly suitable for verification tasks, as evidenced by markedly reduced verification times in practical case studies (especially where many tokens accumulate).

3. Handling Monotonicity-Breaking Features

Unlike classic approaches (working solely for monotonic systems), the method rigorously accounts for monotonicity-breaking features. Age invariants and inhibitor arcs complicate the partial ordering of markings; however, by restricting eligibility for inclusion checks (tokens in untimed places without outgoing inhibitor arcs and whose ages are above relevant bounds), the technique remains sound. Positions sensitive to such features require exact token count or age comparison, precluding reduction for those tokens.

4. Comparison with Translation to Timed Automata (UPPAAL)

Direct DBM-based algorithms with arc reduction outperform translation-based approaches (e.g., translation of TAPN to UPPAAL timed automata) in several respects. Extrapolation and symmetry reduction can be natively incorporated without exponential blow-up, and error traces (precise counterexamples) reflecting transition/fire order and concrete delay are retained—features potentially lost in translation-based workflows. In empirical studies, TAPAAL verified models with the arc reduction technique yield competitive or superior performance and full traceability.

Approach	State-Space Size	Error Trace Fidelity	Overhead
Direct DBM + arc reduction	Reduced	Full	Modest
UPPAAL translation	Larger	Partial (may be lost)	Higher

5. Theoretical Underpinnings and Correctness

The formal definition of the monotonic inclusion relation underpins the correctness theorem: if $M \sqsubseteq M'$ and $M \rightarrow^* M_1$ then there exists $M_1'$ such that $M' \rightarrow^* M_1'$ and $M_1 \sqsubseteq M_1'$. This ensures that reachability is preserved under reduction and allows sound application in model checking. The refined ordering is monotonic on tokens in places that do not break monotonicity, and the procedure for token eligibility guarantees that critical behaviors are never suppressed by the reduction.

6. Limitations and Scope

The additional cost of checking the refined inclusion may not yield savings for 1-safe nets or models with few tokens; the technique produces the largest relative benefit for high-token-count systems. The ordering is defined only for places not involved in equality/upper-bound checks; as such, verification properties that probe for precise token counts (e.g., $p = n$) require disabling arc reduction for the associated places.

7. Practical Implications

The arc reduction approach described in (David et al., 2012) enables efficient verification, reduced memory footprint, and the ability to reliably analyze models with advanced features (transport arcs, inhibitor arcs, age invariants) using forward reachability in symbolic state spaces. Its implementation in TAPAAL provides practitioners with a performant tool that circumvents scalability bottlenecks seen in translation-based workflows and offers error traces suitable for debugging and refinement. The method is particularly well-suited for protocol verification, timed circuits, and workflow systems modeled as bounded TAPNs with timing constraints and advanced arc semantics.